Kramers expression

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

See, for example, Poliak E 1986 Theory of activated rate processes a new derivation of Kramers expression J. Chem. Phys. 85 865... 

The simulation of motion of the solvent as a Langevin-type Brownian motion is a most useful device to reduce computing time. Nevertheless, the essential features of the solvent cage are preserved. Allen was particularly interested in the reaction occurring in the low coupling limit, that is where the solvent does not interact with the particle which is transfered, B. However, when there is some interaction, the rate coefficient becomes diffusion-limited and can be described by a Kramers expression quite well (see Chap. 8, Sect. 2.4) [67]. 

The main problem is the evaluation of the quantity D. Kramers expresses it in terms of the average values of the force acting on the system, the square of the force, the cube of the force and similar expressions whose practical evaluation is very difficult. 

It should be stressed that for the double-well reaction model in the non-Markovian case a general result similar to the Kramers expression (4.160) cannot be found. To evaluate the thermally activated escape rate, the motion within the barrier region is described by means of a GLE in which the potential near the barrier is linearized, that is,... 

The general Kramers expression (6) may be simplified in the region of very high friction, y 2ug, where a purely diffusive regime is attained. 

Though Kramers expression describes a somewhat complicated viscosity dependence and hence is difficult to apply to extract microviscosity, the relation becomes very simple at very high viscosity (Smoluchowski s limit). Since many organized assemblies possess very high microviscosity it is reasonable to assume that the Smoluchowski limit is valid for them. We will demonstrate that assuming the Smoluchowski limit, the microviscosities of some organized assemblies can be estimated quite accurately. 

E. Poliak, J. Chem. Phys., 85, 865 (1986). Theory of Activated Rate Processes A New Derivation of Kramers Expression. 

The simplest model for dilute polymer solutions is to idealize the polymer molecule as an elastic dumbbell consisting of two beads connected by a Hookean spring immersed in a viscous fluid (Fig. 2.1). The spring has an elastic constant Hq. Each bead is associated with a frictional factor C and a negligible mass. If the instantaneous locations of the two beads in space are riand r2, respectively, then the end-to-end vector, R = ri — ri, describes the overall orientation and the internal conformation of the polymer molecule. The polymer-contributed stress tensor can be related to the second-order moment of R. There are two expressions namely the Kramers expression and the Giesekus expression, respectively (Bird et al. 1987b) ... 

For the Hookean spring, the connector force — H t and hence the Kramers expression becomes... 

Fig.2. Viscosity dependence of the rate of activated bairrier crossing, k is the energy controlled rate, kjQ is the rate obtained from Kramers expression, and k5jj is the rate in the Smoluchowski-limit region, (from G.R. Flemming, 1986).

Figure 4. Plot of the pree3qx>nential factor of the isomerization rate vs. viscosity for stilbene in normal alkanes, (o,, x, +) experimental points. The dashed line is a fit to Kramers expression (equation (9)) (from Courtney et al.,1985)...

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